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Properties

Label 12.24.1.k.1
Level $12$
Index $24$
Genus $1$
Analytic rank $0$
Cusps $4$
$\Q$-cusps $4$

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Invariants

Level: $12$ $\SL_2$-level: $12$ Newform level: $24$
Index: $24$ $\PSL_2$-index:$24$
Genus: $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (all of which are rational) Cusp widths $2\cdot4\cdot6\cdot12$ Cusp orbits $1^{4}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 12F1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 12.24.1.5

Level structure

$\GL_2(\Z/12\Z)$-generators: $\begin{bmatrix}5&2\\0&7\end{bmatrix}$, $\begin{bmatrix}5&5\\6&7\end{bmatrix}$, $\begin{bmatrix}11&3\\6&7\end{bmatrix}$, $\begin{bmatrix}11&11\\6&5\end{bmatrix}$
$\GL_2(\Z/12\Z)$-subgroup: $C_2^4.D_6$
Contains $-I$: yes
Quadratic refinements: 12.48.1-12.k.1.1, 12.48.1-12.k.1.2, 12.48.1-12.k.1.3, 12.48.1-12.k.1.4, 12.48.1-12.k.1.5, 12.48.1-12.k.1.6, 12.48.1-12.k.1.7, 12.48.1-12.k.1.8, 24.48.1-12.k.1.1, 24.48.1-12.k.1.2, 24.48.1-12.k.1.3, 24.48.1-12.k.1.4, 24.48.1-12.k.1.5, 24.48.1-12.k.1.6, 24.48.1-12.k.1.7, 24.48.1-12.k.1.8, 24.48.1-12.k.1.9, 24.48.1-12.k.1.10, 24.48.1-12.k.1.11, 24.48.1-12.k.1.12, 24.48.1-12.k.1.13, 24.48.1-12.k.1.14, 24.48.1-12.k.1.15, 24.48.1-12.k.1.16, 60.48.1-12.k.1.1, 60.48.1-12.k.1.2, 60.48.1-12.k.1.3, 60.48.1-12.k.1.4, 60.48.1-12.k.1.5, 60.48.1-12.k.1.6, 60.48.1-12.k.1.7, 60.48.1-12.k.1.8, 84.48.1-12.k.1.1, 84.48.1-12.k.1.2, 84.48.1-12.k.1.3, 84.48.1-12.k.1.4, 84.48.1-12.k.1.5, 84.48.1-12.k.1.6, 84.48.1-12.k.1.7, 84.48.1-12.k.1.8, 120.48.1-12.k.1.1, 120.48.1-12.k.1.2, 120.48.1-12.k.1.3, 120.48.1-12.k.1.4, 120.48.1-12.k.1.5, 120.48.1-12.k.1.6, 120.48.1-12.k.1.7, 120.48.1-12.k.1.8, 120.48.1-12.k.1.9, 120.48.1-12.k.1.10, 120.48.1-12.k.1.11, 120.48.1-12.k.1.12, 120.48.1-12.k.1.13, 120.48.1-12.k.1.14, 120.48.1-12.k.1.15, 120.48.1-12.k.1.16, 132.48.1-12.k.1.1, 132.48.1-12.k.1.2, 132.48.1-12.k.1.3, 132.48.1-12.k.1.4, 132.48.1-12.k.1.5, 132.48.1-12.k.1.6, 132.48.1-12.k.1.7, 132.48.1-12.k.1.8, 156.48.1-12.k.1.1, 156.48.1-12.k.1.2, 156.48.1-12.k.1.3, 156.48.1-12.k.1.4, 156.48.1-12.k.1.5, 156.48.1-12.k.1.6, 156.48.1-12.k.1.7, 156.48.1-12.k.1.8, 168.48.1-12.k.1.1, 168.48.1-12.k.1.2, 168.48.1-12.k.1.3, 168.48.1-12.k.1.4, 168.48.1-12.k.1.5, 168.48.1-12.k.1.6, 168.48.1-12.k.1.7, 168.48.1-12.k.1.8, 168.48.1-12.k.1.9, 168.48.1-12.k.1.10, 168.48.1-12.k.1.11, 168.48.1-12.k.1.12, 168.48.1-12.k.1.13, 168.48.1-12.k.1.14, 168.48.1-12.k.1.15, 168.48.1-12.k.1.16, 204.48.1-12.k.1.1, 204.48.1-12.k.1.2, 204.48.1-12.k.1.3, 204.48.1-12.k.1.4, 204.48.1-12.k.1.5, 204.48.1-12.k.1.6, 204.48.1-12.k.1.7, 204.48.1-12.k.1.8, 228.48.1-12.k.1.1, 228.48.1-12.k.1.2, 228.48.1-12.k.1.3, 228.48.1-12.k.1.4, 228.48.1-12.k.1.5, 228.48.1-12.k.1.6, 228.48.1-12.k.1.7, 228.48.1-12.k.1.8, 264.48.1-12.k.1.1, 264.48.1-12.k.1.2, 264.48.1-12.k.1.3, 264.48.1-12.k.1.4, 264.48.1-12.k.1.5, 264.48.1-12.k.1.6, 264.48.1-12.k.1.7, 264.48.1-12.k.1.8, 264.48.1-12.k.1.9, 264.48.1-12.k.1.10, 264.48.1-12.k.1.11, 264.48.1-12.k.1.12, 264.48.1-12.k.1.13, 264.48.1-12.k.1.14, 264.48.1-12.k.1.15, 264.48.1-12.k.1.16, 276.48.1-12.k.1.1, 276.48.1-12.k.1.2, 276.48.1-12.k.1.3, 276.48.1-12.k.1.4, 276.48.1-12.k.1.5, 276.48.1-12.k.1.6, 276.48.1-12.k.1.7, 276.48.1-12.k.1.8, 312.48.1-12.k.1.1, 312.48.1-12.k.1.2, 312.48.1-12.k.1.3, 312.48.1-12.k.1.4, 312.48.1-12.k.1.5, 312.48.1-12.k.1.6, 312.48.1-12.k.1.7, 312.48.1-12.k.1.8, 312.48.1-12.k.1.9, 312.48.1-12.k.1.10, 312.48.1-12.k.1.11, 312.48.1-12.k.1.12, 312.48.1-12.k.1.13, 312.48.1-12.k.1.14, 312.48.1-12.k.1.15, 312.48.1-12.k.1.16
Cyclic 12-isogeny field degree: $2$
Cyclic 12-torsion field degree: $8$
Full 12-torsion field degree: $192$

Jacobian

Conductor: $2^{3}\cdot3$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 24.2.a.a

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - x^{2} - 24x - 36 $
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Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Weierstrass model
$(-2:0:1)$, $(-3:0:1)$, $(0:1:0)$, $(6:0:1)$

Maps to other modular curves

$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{4x^{2}y^{6}+4623x^{2}y^{4}z^{2}+1573568x^{2}y^{2}z^{4}+167772241x^{2}z^{6}+30xy^{6}z+25278xy^{4}z^{3}+7861513xy^{2}z^{5}+855637692xz^{7}+y^{8}+802y^{6}z^{2}+227959y^{4}z^{4}+26205889y^{2}z^{6}+1056963636z^{8}}{z^{2}y^{2}(6x^{2}y^{2}+x^{2}z^{2}+33xy^{2}z-4xz^{3}+y^{4}+41y^{2}z^{2}-12z^{4})}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
See a full page version of the diagram

The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank Kernel decomposition
$X_0(3)$ $3$ $6$ $6$ $0$ $0$ full Jacobian
4.6.0.d.1 $4$ $4$ $4$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
4.6.0.d.1 $4$ $4$ $4$ $0$ $0$ full Jacobian
$X_0(6)$ $6$ $2$ $2$ $0$ $0$ full Jacobian

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
12.48.1.b.1 $12$ $2$ $2$ $1$ $0$ dimension zero
12.48.1.e.1 $12$ $2$ $2$ $1$ $0$ dimension zero
12.48.1.m.1 $12$ $2$ $2$ $1$ $0$ dimension zero
12.48.1.n.1 $12$ $2$ $2$ $1$ $0$ dimension zero
12.72.3.cy.1 $12$ $3$ $3$ $3$ $0$ $1^{2}$
24.48.1.ct.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.dr.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.ja.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.jd.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.3.a.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.48.3.b.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.48.3.bs.1 $24$ $2$ $2$ $3$ $2$ $1^{2}$
24.48.3.bt.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.48.3.ci.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.48.3.cj.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.48.3.cm.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.48.3.cn.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
36.72.3.w.1 $36$ $3$ $3$ $3$ $0$ $1^{2}$
36.72.5.j.1 $36$ $3$ $3$ $5$ $1$ $1^{4}$
36.72.5.n.1 $36$ $3$ $3$ $5$ $0$ $1^{4}$
60.48.1.bo.1 $60$ $2$ $2$ $1$ $0$ dimension zero
60.48.1.bp.1 $60$ $2$ $2$ $1$ $0$ dimension zero
60.48.1.bs.1 $60$ $2$ $2$ $1$ $0$ dimension zero
60.48.1.bt.1 $60$ $2$ $2$ $1$ $0$ dimension zero
60.120.9.ds.1 $60$ $5$ $5$ $9$ $1$ $1^{8}$
60.144.9.fw.1 $60$ $6$ $6$ $9$ $0$ $1^{8}$
60.240.17.ng.1 $60$ $10$ $10$ $17$ $2$ $1^{16}$
84.48.1.bo.1 $84$ $2$ $2$ $1$ $?$ dimension zero
84.48.1.bp.1 $84$ $2$ $2$ $1$ $?$ dimension zero
84.48.1.bs.1 $84$ $2$ $2$ $1$ $?$ dimension zero
84.48.1.bt.1 $84$ $2$ $2$ $1$ $?$ dimension zero
84.192.13.bd.1 $84$ $8$ $8$ $13$ $?$ not computed
120.48.1.bzk.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.bzn.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.bzw.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.bzz.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.3.cm.1 $120$ $2$ $2$ $3$ $?$ not computed
120.48.3.cn.1 $120$ $2$ $2$ $3$ $?$ not computed
120.48.3.cq.1 $120$ $2$ $2$ $3$ $?$ not computed
120.48.3.cr.1 $120$ $2$ $2$ $3$ $?$ not computed
120.48.3.cy.1 $120$ $2$ $2$ $3$ $?$ not computed
120.48.3.cz.1 $120$ $2$ $2$ $3$ $?$ not computed
120.48.3.dc.1 $120$ $2$ $2$ $3$ $?$ not computed
120.48.3.dd.1 $120$ $2$ $2$ $3$ $?$ not computed
132.48.1.bo.1 $132$ $2$ $2$ $1$ $?$ dimension zero
132.48.1.bp.1 $132$ $2$ $2$ $1$ $?$ dimension zero
132.48.1.bs.1 $132$ $2$ $2$ $1$ $?$ dimension zero
132.48.1.bt.1 $132$ $2$ $2$ $1$ $?$ dimension zero
132.288.21.bb.1 $132$ $12$ $12$ $21$ $?$ not computed
156.48.1.bo.1 $156$ $2$ $2$ $1$ $?$ dimension zero
156.48.1.bp.1 $156$ $2$ $2$ $1$ $?$ dimension zero
156.48.1.bs.1 $156$ $2$ $2$ $1$ $?$ dimension zero
156.48.1.bt.1 $156$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.bzi.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.bzl.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.bzu.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.bzx.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.3.by.1 $168$ $2$ $2$ $3$ $?$ not computed
168.48.3.bz.1 $168$ $2$ $2$ $3$ $?$ not computed
168.48.3.cc.1 $168$ $2$ $2$ $3$ $?$ not computed
168.48.3.cd.1 $168$ $2$ $2$ $3$ $?$ not computed
168.48.3.cg.1 $168$ $2$ $2$ $3$ $?$ not computed
168.48.3.ch.1 $168$ $2$ $2$ $3$ $?$ not computed
168.48.3.ck.1 $168$ $2$ $2$ $3$ $?$ not computed
168.48.3.cl.1 $168$ $2$ $2$ $3$ $?$ not computed
204.48.1.bo.1 $204$ $2$ $2$ $1$ $?$ dimension zero
204.48.1.bp.1 $204$ $2$ $2$ $1$ $?$ dimension zero
204.48.1.bs.1 $204$ $2$ $2$ $1$ $?$ dimension zero
204.48.1.bt.1 $204$ $2$ $2$ $1$ $?$ dimension zero
228.48.1.bo.1 $228$ $2$ $2$ $1$ $?$ dimension zero
228.48.1.bp.1 $228$ $2$ $2$ $1$ $?$ dimension zero
228.48.1.bs.1 $228$ $2$ $2$ $1$ $?$ dimension zero
228.48.1.bt.1 $228$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.bzi.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.bzl.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.bzu.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.bzx.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.3.by.1 $264$ $2$ $2$ $3$ $?$ not computed
264.48.3.bz.1 $264$ $2$ $2$ $3$ $?$ not computed
264.48.3.cc.1 $264$ $2$ $2$ $3$ $?$ not computed
264.48.3.cd.1 $264$ $2$ $2$ $3$ $?$ not computed
264.48.3.cg.1 $264$ $2$ $2$ $3$ $?$ not computed
264.48.3.ch.1 $264$ $2$ $2$ $3$ $?$ not computed
264.48.3.ck.1 $264$ $2$ $2$ $3$ $?$ not computed
264.48.3.cl.1 $264$ $2$ $2$ $3$ $?$ not computed
276.48.1.bo.1 $276$ $2$ $2$ $1$ $?$ dimension zero
276.48.1.bp.1 $276$ $2$ $2$ $1$ $?$ dimension zero
276.48.1.bs.1 $276$ $2$ $2$ $1$ $?$ dimension zero
276.48.1.bt.1 $276$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.bzk.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.bzn.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.bzw.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.bzz.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.3.by.1 $312$ $2$ $2$ $3$ $?$ not computed
312.48.3.bz.1 $312$ $2$ $2$ $3$ $?$ not computed
312.48.3.cc.1 $312$ $2$ $2$ $3$ $?$ not computed
312.48.3.cd.1 $312$ $2$ $2$ $3$ $?$ not computed
312.48.3.cg.1 $312$ $2$ $2$ $3$ $?$ not computed
312.48.3.ch.1 $312$ $2$ $2$ $3$ $?$ not computed
312.48.3.ck.1 $312$ $2$ $2$ $3$ $?$ not computed
312.48.3.cl.1 $312$ $2$ $2$ $3$ $?$ not computed